Contents

  1. Part I: RJMCMC - \(exp(1)\) prior on \(\beta\)
  1. Data
  1. RJMCMC
  1. Part II: RJMCMC - \(\Gamma(k, \theta)\) prior on \(\beta\)

Background

RJMCMC was implemented whereby the algorithm jumped between the Baseline model \(M_1\) of an epidemic with regular spreading events \((\alpha)\) and the Super Spreading Events (SSE) model \((\alpha, \beta, \gamma)\) which has both regular spreading events with rate \(\alpha\) and super spreading events with rate \(\beta\) and multiplicative factor \(\gamma\).

Bayes Factor

Bayesian model comparison is a method of model selection based on Bayes factors. The aim of the Bayes factor is to quantify the support for one model over another, e.g model \(M_1\) over model \(M_2\). The Bayes Factor BF is as follows;
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)P(D)}{P(M_1)}}{\dfrac{P(M_2|D)P(D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_2 | D)} \]
when \(P(M_1) == P(M_2)\), otherwise
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)}{P(M_1)}}{\dfrac{P(M_2|D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_1)} \cdot \dfrac{P(M_2)}{P(M_2|D)} \]
where \(P(D|M_1)\) is the model evidence, specifically the marginal likelihood integrand;
\[ P(D|M_1) = \int P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta) \hspace{1 mm} P(\hspace{1 mm}\theta \hspace{1 mm}| M_1) \hspace{1 mm}d \theta\] and the first term in the integrand \(P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta)\) is the likelihood and the second term \(P(\hspace{1 mm}\theta \hspace{1 mm}| M_1)\) is the prior on the model parameter \(\theta\).



Interpretation of the Bayes Factor results


A Bayes Factor > 1 signifies that \(M_1\) is more strongly supported by the data under consideration than \(M_2\). Harold Jefferys gave a scale of interpretation of the Bayes Factor;

Bayes Factor Bayes Factor equivalence Evidence Strength
< 10^0 < 1 Negative (supports M_2)
[10^0, 10^1/2] [1, 3.16] Weak evidence
[10^1/2, 10^1] [3.16, 10] Substantial
[10^1, 10^3/2] [10, 31.62] Strong
[10^3/2, 10^2] [31.62, 100] Very strong
> 10^2 > 100 Decisive


Part I: RJMCMC between Base model & SSE Model - exp(1) prior used for beta

An \(exp\hspace{1mm}(\beta; \hspace{1mm}1)\) has density;

Model Comparison - SSE data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.32  0.1 0.01    10 0.18 1.8  0.23          70.6   86.57
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        941   12.33        134     99.45         1081          843
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          244     77.55          842         8070      9.45   0.8913
##   beta_pc_non_0  bf
## 1        0.1087 8.2


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.66  0.1 0.05    10 4.36 1.8  0.91          1.94    4.35
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        435     1.8        180       4.2          420            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.45  0.1 0.15    10 6.53 1.8   1.4           5.2    6.16
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        616    2.01        201      6.15          615            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.78  0.1 0.02    10 0.26 1.8  0.82         10.43   83.89
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1286   14.29        219     93.41         1432         1435
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           98     93.61         1434         7032     16.94   0.8467
##   beta_pc_non_0    bf
## 1        0.1533 5.523


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.75  0.1 0.02    10 0.18 1.8  1.77          3.66   54.12
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        657    8.81        107      42.5          516         1202
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           12     99.01         1201         7584     13.67   0.8786
##   beta_pc_non_0    bf
## 1        0.1214 7.237


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.73  0.1 0.08    10 0.45 1.8  1.85         10.17   62.97
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1779    9.98        282     40.25         1137         2600
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          225     92.04         2599         4575     36.23   0.7175
##   beta_pc_non_0   bf
## 1        0.2825 2.54

Part 2: Gamma Prior on Beta

A gamma prior, \(\Gamma(\beta; k, \theta)\) on beta was also trialed whereby \(k\) determines the shape of the distribution and \(\theta\) governs the scale. The gamma distribution function is as follows;

\[ \Gamma(\beta; k, \theta) = \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \]

A range of gamma priors on beta are used including a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 2)\), \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 3, \hspace{1mm} 3)\), \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 4, \hspace{1mm} 4)\) and \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 5, \hspace{1mm} 5)\). Each have a mean of \(k \cdot \theta\). A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), is as follows;


A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 2)\);



And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 3)\);

And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}4, \hspace{1mm} 4)\);

And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}5, \hspace{1mm} 5)\);



In the Metropolis acceptance step, the logs of all quantities are determined and evaluating \(log\hspace{1mm}( \Gamma\hspace{1mm}(\beta; \hspace{1mm} k, \hspace{1mm} \theta))\) gives;


\[log\bigg( \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \bigg)\]

\[ = \dfrac{1}{log\Gamma(k)\cdot klog(\theta)} \cdot (k-1) \cdot log(\beta) \cdot \dfrac{-\beta}{\theta} \]


Gamma Prior \(\Gamma\)(\(\beta\); 2, 2.5)


Model Comparison - SSE data I


##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.38  0.1 0.04    10 0.33 1.8  0.47         71.41       0
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          0   16.37        292      0.22            4         1330
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          454     74.55         1329         6886     16.18   0.8216
##   beta_pc_non_0    bf
## 1        0.1784 4.605


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.16  0.1  0.5    10 3.74 1.8  2.02         55.09       0
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          0   14.81       1481         0            0            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.07  0.1  0.5    10 3.73 1.8  1.93         25.29       0
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          0    3.54        354         0            0            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.82  0.1 0.03    10 0.28 1.8  0.87         10.82    0.06
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          1   13.55        212      0.77           12         1522
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           42     97.31         1522         6913     18.04   0.8435
##   beta_pc_non_0   bf
## 1        0.1565 5.39


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.75  0.1 0.02    10 0.19 1.8  1.78          3.29       0
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          0    9.82        119      0.41            5         1206
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1            6      99.5         1205         7582     13.71   0.8788
##   beta_pc_non_0    bf
## 1        0.1212 7.251


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.73  0.1 0.08    10 0.45 1.8  1.85          9.44    0.18
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          5   10.16        282      2.27           63         2577
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          199     92.83         2577         4646     35.68   0.7223
##   beta_pc_non_0    bf
## 1        0.2777 2.601


Gamma Prior \(\Gamma(\beta; 3, 2)\)


Model Comparison - SSE data I


##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.32  0.1 0.02    10 0.26 1.8  0.32          71.3    95.7
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1335   13.05        182     88.03         1228         1026
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          369     73.55         1025         7579     11.91   0.8605
##   beta_pc_non_0    bf
## 1        0.1395 6.168


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.53  0.1 0.08    10 4.91 1.8  0.96          3.37    7.05
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        705    3.24        324      6.79          679            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.59  0.1 0.12    10  6.1 1.8  1.36          4.36    4.18
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        418    1.63        163      4.14          414            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.81  0.1 0.03    10 0.26 1.8  0.85         10.31   90.29
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1395    13.4        207     79.48         1228         1438
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          107     93.07         1437         7017        17   0.8455
##   beta_pc_non_0    bf
## 1        0.1545 5.472


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.76  0.1 0.02    10 0.19 1.8  1.78          3.73    55.3
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        689    9.63        120     36.76          458         1232
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           14     98.88         1231         7522     14.06   0.8754
##   beta_pc_non_0    bf
## 1        0.1246 7.026


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.73  0.1 0.09    10 0.48 1.8  1.87         10.16   61.81
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1772     9.7        278     31.88          914         2608
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          259     90.97         2607         4525     36.55   0.7133
##   beta_pc_non_0    bf
## 1        0.2867 2.488


Gamma Prior \(\Gamma(\beta; 3, 3)\)


Model Comparison - SSE data I


##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.32  0.1 0.02    10 0.26 1.8  0.32          71.3    95.7
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1335   13.05        182     88.03         1228         1026
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          369     73.55         1025         7579     11.91   0.8605
##   beta_pc_non_0    bf
## 1        0.1395 6.168


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.53  0.1 0.08    10 4.91 1.8  0.96          3.37    7.05
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        705    3.24        324      6.79          679            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.59  0.1 0.12    10  6.1 1.8  1.36          4.36    4.18
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        418    1.63        163      4.14          414            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.81  0.1 0.03    10 0.26 1.8  0.85         10.31   90.29
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1395    13.4        207     79.48         1228         1438
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          107     93.07         1437         7017        17   0.8455
##   beta_pc_non_0    bf
## 1        0.1545 5.472


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.76  0.1 0.02    10 0.19 1.8  1.78          3.73    55.3
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        689    9.63        120     36.76          458         1232
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           14     98.88         1231         7522     14.06   0.8754
##   beta_pc_non_0    bf
## 1        0.1246 7.026


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.73  0.1 0.09    10 0.48 1.8  1.87         10.16   61.81
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1772     9.7        278     31.88          914         2608
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          259     90.97         2607         4525     36.55   0.7133
##   beta_pc_non_0    bf
## 1        0.2867 2.488


Gamma Prior \(\Gamma(\beta; 4, 4)\)


Model Comparison - SSE data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.31  0.1 0.02    10 0.27 1.8  0.34         71.28   95.47
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1390    13.6        198     86.68         1262         1077
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          379     73.97         1076         7467      12.6   0.8544
##   beta_pc_non_0    bf
## 1        0.1456 5.868


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.73  0.1 0.07    10 5.08 1.8  1.15          2.62    6.05
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        605    2.74        274      5.84          584            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.55  0.1 0.14    10 7.25 1.8  1.53          7.51    7.71
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        771       3        300      8.09          809            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.79  0.1 0.02    10 0.27 1.8  0.83         10.55   91.21
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1411   13.57        210     76.08         1177         1444
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          103     93.34         1443         7009     17.07   0.8453
##   beta_pc_non_0    bf
## 1        0.1547 5.464


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.76  0.1 0.02    10 0.19 1.8  1.79          3.73   53.71
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        680    9.72        123     35.23          446         1256
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           10     99.21         1255         7478     14.37   0.8734
##   beta_pc_non_0    bf
## 1        0.1266 6.899


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.72  0.1 0.09    10 0.48 1.8  1.87         10.05   61.86
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1766    9.42        269     31.38          896         2576
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          279     90.23         2575         4569     36.04   0.7145
##   beta_pc_non_0    bf
## 1        0.2855 2.503



Gamma Prior \(\Gamma(\beta; 5, 5)\)


Model Comparison - SSE data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.32  0.1 0.02    10 0.24 1.8  0.31         71.28   95.98
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1314   12.49        171     88.24         1208         1019
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          350     74.43         1018         7612      11.8   0.8631
##   beta_pc_non_0    bf
## 1        0.1369 6.305


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.53  0.1 0.08    10 4.91 1.8  0.96          3.37    7.05
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        705    3.24        324      6.79          679            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.59  0.1 0.12    10  6.1 1.8  1.36          4.74    4.99
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        499    1.79        179       4.9          490            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8  0.8  0.1 0.03    10 0.27 1.8  0.85          10.5   90.63
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1422   13.19        207     77.12         1210         1477
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           92     94.14         1477         6953     17.52    0.843
##   beta_pc_non_0    bf
## 1         0.157 5.369


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.76  0.1 0.02    10  0.2 1.8  1.78          3.85   54.43
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        682    9.34        117     36.55          458         1241
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           12     99.04         1240         7506     14.18   0.8747
##   beta_pc_non_0    bf
## 1        0.1253 6.981


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.72  0.1  0.1    10  0.5 1.8  1.88         10.39   59.66
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1739    9.26        270     31.94          931         2599
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          316     89.16         2598         4486     36.67   0.7085
##   beta_pc_non_0    bf
## 1        0.2915 2.431


Bayes Factor - Summary of Results

In this particular setting in which the Baseline model \(\alpha\) and SSE model \((\alpha, \beta, \gamma)\) are compared, the Bayes factor is calculated as;

\[ \dfrac{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples == 0}{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples != 0} \]


The following table summaries the results of the RJMCMC iterations for a number of datasets when both a \(exp(\beta, 1)\) prior and a variety of \(\Gamma(\beta;)\) priors on beta were used.



Epidemic Data Max daily infection count Bayes Factor exp(beta; 1) Bayes Factor,gamma(2, 2.5) Bayes Factor,gamma(3, 2) Bayes Factor,gamma(3, 3) Bayes Factor,gamma(4, 4) Bayes Factor,gamma(5, 5)
SSE - dies out 2 8.2 4.605 6.168 6.168 5.868 6.305
SSE spreads 55 0 0 0 0 0 0
SS spreads 340 0 0 0 0 0 0
SSE - dies out 2 5.523 5.39 5.472 5.472 5.464 5.369
Base - spreads 65 7.237 7.251 7.026 7.026 6.899 6.981
Base - spreads 45 2.54 2.601 2.488 2.488 2.503 2.431